Will iMac 27 in (year 2013) be able to use as a display for Mac Studio M3 Ultra?

+1 No iMacs are computers, they are not external displays for any other devices.

in short you need to get an external display, there are scores to choose from from about $80-$1500.

No. Target Display Mode is dead and none of the new Mac's support using an old iMac as a display.

Use your iMac as a display with target display mode - Apple Support

If you get a new Mac Studio or Mac mini, you will need to use an external display.

Connect a display to Mac Studio - Apple Support

Connect a display to Mac mini - Apple Support

No. That iMac supports Target Display Mode, using Thunderbolt input – but Apple added restrictions such that no current Mac supports using ANY iMac as a Target Display.

Reference: Use your iMac as a display with target display mode - Apple Support

That iMac is too old to be an AirPlay Receiver for AirPlay to Mac, a lower-quality way of using a Mac as a display.

Reference: Continuity features and requirements for Apple devices - Apple Support

purchase a Studio display and you will never have compatibility issues and have great sound and video, and has a camera for zoom meetings. It is possible to convert an iMac to a monitor by removing the electronics and buying new electronics to drive the LED display, but it's messy and the rear connections don't fit the shell as a built unit does; its not an expensive solution, but if you are affording a MacStudio, also go first class on the monitor. You will be glad you did.

No. Target Display is not supported on the Mac Studio.

You can buy a new good quality 4k display at a fraction of the super overpriced Studio Display.

Dell 27 Plus 4K Monitor S2725QS 27-inch (3840 x 2160) 120Hz, IPS Panel, 16 9 Display - Amazon.com

Samsung 27” S80UA 4K UHD IPS HDR10 High Resolution Monitor - Amazon.com

SAMSUNG 32" ViewFinity S7 (S70D) 4K UHD IPS HDR10 High Resolution Monitor - Amazon.com

LG Ultra Fine 27US500-W 27-Inch 4K UHD - Amazon.com